PEIRCE. — BALLISTIC GALVANOMETERS OF LONG PERIOD. 295 



When a galvanometer is critically damped /3 2 = a 2 , p = 0, and the 



equation of motion is 



,72/3 jq 



% + >•% + »=°> < 18 > 



and the general solution of this is 



6 = (A + Bt) e~ at . (19) 



If when t = 0, = 0\ and w = a/ ; 



e = [6'+ (to' + aff) t] e' at . (20) 



Figure F. 



If when the coil is at rest in its position of equilibrium, an impulsive 

 current sent through the instrument gives the coil an initial angular 

 velocity w , 



= w -t- e~ at , a> = <»o- e~ at (1 - at), (21) 



and if after this motion has gone on until the time t x a second impulse 

 increases the angular velocity by the amount w 2 , then after the second 

 impulse 



$ = u> t e~ at + w 2 (t - h) e~ a (<-*>. (22) 



It is possible to give to this equation also a graphical treatment 



